Unformatted text preview: (b) Suppose ( a ) = ( b ). Then a 2 = b 2 and so ( ab ) 2 = a 22 abb 2 + 2 b 2 = a 2b 2 = 0 Since D has no zero divisors and ( ab ) 2 = 0, it follows that ab = 0 and so a = b . Variations are possible. For example, a 2 = b 2 gives us 0 = a 2b 2 = ( a + b )( ab ) and so the lack of zero divisors gives us a = b . However,x = x2 x = x and so a = b . (c) The degree of ( a ) is always even, hence no polynomials of odd degree are in the image. Aside: In fact the image is precisely Z 2 [ x 2 ] because, as you should be able to prove), ( p ( x )) = p ( x ) 2 = p ( x 2 )....
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 Spring '08
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 Algebra, Addition, Integers, Normal subgroup, Complex number, Integral domain

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